• Title, Summary, Keyword: Riccati Equation

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PROPERTIES ON q-DIFFERENCE RICCATI EQUATION

  • Huang, Zhi-Bo;Zhang, Ran-Ran
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.6
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    • pp.1755-1771
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    • 2018
  • In this paper, we investigate a certain type of q-difference Riccati equation in the complex plane. We prove that q-difference Riccati equation possesses a one parameter family of meromorphic solutions if it has three distinct meromorphic solutions. Furthermore, we find that all meromorphic solutions of q-difference Riccati equation and corresponding second order linear q-difference equation can be expressed by q-gamma function if this q-difference Riccati equation admits two distinct rational solutions and $q{\in}{\mathbb{C}}$ such that 0 < ${\mid}q{\mid}$ < 1. The growth and value distribution of differences of meromorphic solutions of q-difference Riccati equation are also treated.

RICCATI EQUATION IN QUADRATIC OPTIMAL CONTROL PROBLEM OF DAMPED SECOND ORDER SYSTEM

  • Ha, Junhong;Nakagiri, Shin-Ichi
    • Journal of the Korean Mathematical Society
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    • v.50 no.1
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    • pp.173-187
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    • 2013
  • This paper studies the properties of solutions of the Riccati equation arising from the quadratic optimal control problem of the general damped second order system. Using the semigroup theory, we establish the weak differential characterization of the Riccati equation for a general class of the second order distributed systems with arbitrary damping terms.

THE ($\frac{G'}{G}$)- EXPANSION METHOD COMBINED WITH THE RICCATI EQUATION FOR FINDING EXACT SOLUTIONS OF NONLINEAR PDES

  • Zayed, E.M.E.
    • Journal of applied mathematics & informatics
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    • v.29 no.1_2
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    • pp.351-367
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    • 2011
  • In this article, we construct exact traveling wave solutions for nonlinear PDEs in mathematical physics via the (1+1)- dimensional combined Korteweg- de Vries and modified Korteweg- de Vries (KdV-mKdV) equation, the (1+1)- dimensional compouned Korteweg- de Vries Burgers (KdVB) equation, the (2+1)- dimensional cubic Klien- Gordon (cKG) equation, the Generalized Zakharov- Kuznetsov- Bonjanmin- Bona Mahony (GZK-BBM) equation and the modified Korteweg- de Vries - Zakharov- Kuznetsov (mKdV-ZK) equation, by using the (($\frac{G'}{G}$) -expansion method combined with the Riccati equation, where G = $G({\xi})$ satisfies the Riccati equation $G'({\xi})=A+BG^2$ and A, B are arbitrary constants.

Numerical Solution of Riccati Differential Equation in Optimal Control Theory (최적제어이론과 관련된 "리카티" 미분방정식의 수식해)

  • 경규학
    • Journal of the Korean Operations Research and Management Science Society
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    • v.9 no.2
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    • pp.28-33
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    • 1984
  • In this paper some procedures are given whereby an analytic solution may be found for the Riccati differential equation and algebraic Riccati equation in optimal control theory. Some iterative techniques for solving these equations are presented. Rate of convergence and initialization of the iterative processes are discussed.

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THE RECURSIVE ALGOFITHM FOR OPTIMAL REGULATOR OF NONSTANCARD SINGULARLY PERTURVED SYSTEMS

  • Mukaidani, Hiroaki;Xu, Hau;Mizukami, Koichi
    • 제어로봇시스템학회:학술대회논문집
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    • pp.10-13
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    • 1995
  • This paper considers the linear-quadratic optimal regulator problem for nonstandard singularly perturbed systems making use of the recursive technique. We first derive a generalized Riccati differential equation by the Hamilton-Jacobi equation. In order to obtain the feedback gain, we must solve the generalized algebraic Riccati equation. Using the recursive technique, we show that the solution of the generalized algebraic Riccati equation converges with the rate of convergence of O(.epsilon.). The existence of a bounded solution of error term can be proved by the implicit function theorem. It is enough to show that the corresponding Jacobian matrix is nonsingular at .epsilon. = 0. As a result, the solution of optimal regulator problem for nonstandard singularly perturbed systems can be obtained with an accuracy of O(.epsilon.$^{k}$ ). The proposed technique represents a significant improvement since the existing method for the standard singularly perturbed systems can not be applied to the nonstandard singularly perturbed systems.

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Linear Quadratic Regulators with Two-point Boundary Riccati Equations (양단 경계 조건이 있는 리카티 식을 가진 선형 레규레이터)

  • Kwon, Wook-Hyun
    • Journal of the Korean Institute of Telematics and Electronics
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    • v.16 no.5
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    • pp.18-26
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    • 1979
  • This paper extends some well-known system theories on algebraic matrix Lyapunov and Riccati equations. These extended results contain two point boundary conditions in matrix differential equations and include conventional results as special cases. Necessary and sufficient conditions are derived under which linear systems are stabilizable with feedback gains derived from periodic two-point boundary matrix differential equations. An iterative computation method for two-point boundary differential Riccati equations is given with an initial guess method. The results in this paper are related to periodic feedback controls and also to the quadratic cost problem with a discrete state penalty.

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Nonlinear Observer Design for Satellite Angular Rate Estimation by SDRE Method (SDRE 기법을 이용한 위성 각속도 추정용 비선형 관측기 설계)

  • Jin, Jaehyun
    • Journal of the Korean Society for Aeronautical & Space Sciences
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    • v.42 no.10
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    • pp.816-822
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    • 2014
  • The estimation of the angular rate of a satellite has been discussed. A nonlinear observer has been proposed based on the state-dependent Riccati equation method. A sufficient stability condition for the convergence of estimation error has been presented. This condition is related to a state-dependent algebraic Riccati equation. It has been derived by transforming nonlinear error dynamics into a Lipschitz nonlinearity. An observer gain is obtained from this condition. Numerical simulations are presented to verify the proposed method.

Solvability of Stochastic Discrete Algebraic Riccati Equation

  • Oh, Kyu-Kwon;Okuyama, Yoshifumi
    • 제어로봇시스템학회:학술대회논문집
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    • pp.33.4-33
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    • 2001
  • This paper considers a stochastic discrete algebraic Riccati equation, which is a generalized version of the well-known standard discrete algebraic Riccati equation, and has additional linear terms. Under controllability, observability and the assumption that the additional terms are not too large, the existence of a positive definite solution is guaranteed. It is shown that it arises in optimal control of a linear discrete-time system with multiplicative White noise and quadratic cost. A numerical example is given.

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New Upper Matrix Bounds for the Solution of the Continuous Algebraic Riccati Matrix Equation

  • Davies, Richard Keith;Shi, Peng;Wiltshire, Ron
    • International Journal of Control, Automation, and Systems
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    • v.6 no.5
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    • pp.776-784
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    • 2008
  • In this paper, new upper matrix bounds for the solution of the continuous algebraic Riccati equation (CARE) are derived. Following the derivation of each bound, iterative algorithms are developed for obtaining sharper solution estimates. These bounds improve the restriction of the results proposed in a previous paper, and are more general. The proposed bounds are always calculated if the stabilizing solution of the CARE exists. Finally, numerical examples are given to demonstrate the effectiveness of the present schemes.

Composite Control for Weakly Coupled Bilinear Systems with Successive Galerkin Approximation (연속적 Galerkin 근사를 이용한 정규 섭동 쌍일차 시스템에 대한 합성 제어)

  • Kim, Young-Joong;Kim, Beom-Soo;Lim, Myo-Taeg
    • Proceedings of the KIEE Conference
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    • pp.1996-1998
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    • 2001
  • This paper presents the closed-loop composite control for weakly coupled bilinear systems with a quadratic performance criterion. The Riccati equation for weakly coupled bilinear system is decomposed into three reduced Riccati equations by the weak coupling theory, and we obtain optimal solutions of each reduced Riccati equation using successive Galerkin approximation(SGA). We design the composite control law that consists of optimal solutions of each reduced Riccati equation. The proposed algorithm reduces the disadvantages of SGA method.

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